No. A sucker is someone who's an easy mark, credulous, easy to trick. A suck-up is someone who sucks up a lot. To explain further...
If you wanted to construct a noun in the same manner as sucker from the verb suck up, you could plausibly use sucker-up or suck-upper, but that would be a new coinage, not a term that's actually used. A person who sucks is a sucker; a person who sucks up is a suck-up. However, while a sucker can be a person who sucks, that isn't what we usually use that word for.
You have to understand the difference between the basic verb to suck and the phrasal verb to suck up. Sucking up rarely involves any literal sucking. It's an expression to refer to things like sycophancy. Trying to make someone in authority like you or favour you.
The one point of confusion you can end up with here is that while you can suck up to somebody, and if suck up has no object or complement or prepositional phrase indicating who or what it's done to then that can be taken as implicit. But you can also suck up a drink, say, through a straw. In that case, the verb is just to suck and the up is effectively an adverb.
So, that's suck up. What about sucker?
As I mentioned previously, it is true to say that someone who sucks (as in the physical action, rather than the insult) is a sucker, and it occasionally gets used that way, but usually it's used for something else. It's a descriptive noun that means a person who is easily tricked, an "easy mark", someone who is credulous and easy to deceive. It is, in a sense, a particularly specific sort of stupidity.
It also has use as a relatively meaningless term for something you find or have found annoying. That is a distinct meaning from the sense of credulity.